Discretization Independence: Theory & Applications

• Discretization independence is the property that discrete representations preserve the underlying object, ensuring invariance under refinement, coarsening, or mesh changes.
• Methodologies include perfect composition in reparametrization-invariant path integrals and triangulation independence in discrete gravity through compatible measures.
• Practical implications address nonlocality in 4D gravity, bias in discretized statistical tests, and mesh-agnostic performance in operator learning models.

Discretization independence denotes the property that a discrete representation does not change the object being computed, inferred, or approximated. In the cited literature, the term is used in several non-equivalent but structurally related senses: exact invariance under refinement or coarse graining in path integrals, triangulation independence under Pachner moves in discrete gravity, preservation of latent independence relations when only discretized observations are available, asymptotic equivalence between grid-based and function-space dependence measures, and mesh-agnostic behavior in operator learning architectures (Steinhaus, 2011, Dittrich et al., 2011, Hauck et al., 9 Jul 2025). The common theme is that discretization should be an implementation device rather than a source of spurious dynamics, spurious dependence, or representation-specific bias.

Setting	What is required to survive discretization	Canonical criterion
Reparametrization-invariant path integrals	continuum dynamics	$K = K \circ K$
Discrete gravity	retriangulation of the bulk	Pachner-move invariance
CI testing with discretized observations	latent covariance or precision structure	bridge equations, GMM
Functional dependence measures	independence characterization	same asymptotic target
Operator learning	changes of mesh, basis, or fidelity	large effective discretization set

1. Exact composition and perfect discretization

In the path-integral setting, discretization independence is formulated most sharply for reparametrization-invariant systems. Starting from an ordinary action

$S=\int dt\,L(q,\dot q),$

one introduces an auxiliary parameter $s$ and treats $t=t(s)$ as dynamical, obtaining

$$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$

This form is invariant under reparametrizations $q(s),t(s)\to q(f(s)),t(f(s))$. In canonical language the symmetry appears as the constraint

$\tilde H = p_t + H = 0.$

A naïve discretization replaces $q(s),t(s)$ by lattice variables $q_n,t_n$ and derivatives by finite differences. The resulting lattice action generically breaks reparametrization symmetry because the subdivision itself selects preferred vertex positions; broken gauge modes then become physical. In this framework, broken symmetry and discretization dependence are presented as equivalent problems (Steinhaus, 2011).

The remedy is an iterative improvement procedure of Wilsonian type. Given a discretized propagator $K^{(n)}$, one integrates out an internal vertex,

$S=\int dt\,L(q,\dot q),$0

and interprets the resulting flow as a renormalization-group map. Fixed points of this map are “perfect discretizations”: amplitudes invariant under further coarse graining and therefore exact on the lattice. For generalized Gaussian ansätze, the paper derives explicit recursion relations, such as

$S=\int dt\,L(q,\dot q),$1

and emphasizes that convergence depends crucially on the short-time measure behavior

$S=\int dt\,L(q,\dot q),$2

The central identity is the perfect-composition law

$S=\int dt\,L(q,\dot q),$3

obtained when the discrete propagator is invariant under vertex translations, the lattice analogue of reparametrization invariance. This identity means that inserting or removing intermediate lattice points does not alter the amplitude. In that precise sense, reparametrization invariance implies discretization independence. Because repeated composition does nothing beyond the first application, the path integral acts as a projector onto physical states satisfying $S=\int dt\,L(q,\dot q),$4, rather than as ordinary evolution in an external time parameter (Steinhaus, 2011).

2. Triangulation independence in discrete gravity

In discrete gravity, discretization independence is usually formulated as triangulation independence. Linearized Regge calculus provides the canonical test bed: the continuum metric is replaced by edge lengths on a triangulation, and one asks whether amplitudes are invariant under local retriangulations generated by Pachner moves. Around flat backgrounds, where remnants of discrete diffeomorphism symmetry survive, this requirement becomes technically accessible. The analysis expands edge lengths as

$S=\int dt\,L(q,\dot q),$5

and the Regge action to quadratic order about a flat solution, so that the Hessian controls the Gaussian path integral (Dittrich et al., 2011).

A key point is that the action alone is insufficient. One must also choose a compatible path-integral measure. For a local ansatz, the paper studies measures of the form

$S=\int dt\,L(q,\dot q),$6

and derives factorized Hessians for Pachner moves in both three and four dimensions. In three dimensions this leads to a local measure

$S=\int dt\,L(q,\dot q),$7

which exactly cancels Pachner-move dependence in linearized Regge calculus. The result is triangulation independence under the $S=\int dt\,L(q,\dot q),$8 and $S=\int dt\,L(q,\dot q),$9 moves, after gauge fixing the vertex-displacement zero modes in the latter case. The same factor coincides with the semiclassical asymptotics of the Ponzano–Regge model, so the measure is not merely ad hoc but matches a known 3D spin-foam normalization (Dittrich et al., 2011).

In four dimensions, the corresponding local measure is

$s$0

and it yields the correct local transformation behavior for the $s$1 and $s$2 moves only up to an additional nonlocal factor $s$3. Full triangulation independence fails more fundamentally because the classical Regge action is not invariant under the $s$4 move. The special condition

$s$5

would be needed for a local measure of this form to be $s$6 invariant, and that condition is not generic on flat backgrounds. The 3D/4D contrast is therefore sharp: exact triangulation independence is realized in the linearized 3D theory, while in 4D it is at best partial (Dittrich et al., 2011).

3. Four-dimensional obstructions and non-locality

The 4D obstruction can be stated more strongly: no local path-integral measure makes the 4D linearized Regge theory invariant under the $s$7 and $s$8 Pachner moves. The obstruction is encoded in an extra factor $s$9 in the Hessian of the Regge action. For edge pairs $t=t(s)$0 and $t=t(s)$1 the Hessian has the structure

$t=t(s)$2

with

$t=t(s)$3

and $t=t(s)$4 in fact independent of the chosen pair $t=t(s)$5 (Dittrich et al., 2014).

The geometric interpretation is decisive. Using the Cayley–Menger matrix of the six vertices involved in the Pachner move, the paper shows that

$t=t(s)$6

where $t=t(s)$7 is the relevant Cayley–Menger minor. In four dimensions, $t=t(s)$8 exactly when the six vertices lie on a common 3-sphere. This condition is global: it depends on the entire six-vertex configuration and cannot be reconstructed from local simplex data. That is why $t=t(s)$9 cannot factorize over 4-simplices or lower-dimensional subsimplices. If it could, the vanishing locus of $$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$0 would be determined locally, contradicting the sphericality criterion (Dittrich et al., 2014).

This yields a precise sense in which discretization independence forces non-locality in 4D. The “almost invariant” local factor

$$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$1

cannot absorb the Hessian determinant in general. A nonlocal correction can be written for the $$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$2 move,

$$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$3

but the compensating factor $$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$4 depends on the entire configuration. A plausible implication is that exact 4D triangulation independence cannot be achieved within purely local length-variable measures in linearized Regge calculus, and must instead involve nonlocality or a reformulation in different variables or boundary-amplitude language (Dittrich et al., 2014).

4. Preservation of independence structures under discretization

Outside gravity, discretization independence often means preservation of an independence structure under a change of representation. One explicit example is ultra-discretization. For a measurable bijection $$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$5, the independence preserving property requires the existence of non-Dirac probability measures $$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$6 such that

$$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$7

With the logarithmic scaling

$$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$8

the paper proves that both the Yang–Baxter property and the independence preserving property are stable under ultra-discretization, provided the maps converge uniformly on compact sets and the pushed-forward measures converge weakly. Quadrirational Yang–Baxter families therefore remain reversible Yang–Baxter maps after tropicalization, and their associated independence-preserving laws pass to mixed exponential limits such as $$S\big(q(s),t(s)\big)= \int ds \, L\Big(q, \frac{q'}{t'}\Big) t' .$$9, $q(s),t(s)\to q(f(s)),t(f(s))$0, and $q(s),t(s)\to q(f(s)),t(f(s))$1 (Kondo et al., 30 Apr 2025).

A second line of work concerns conditional independence testing under discretized observations. Here the relevant issue is not invariance of the observed variables themselves but recovery of the latent target relation. Under the latent Gaussian model, one can have

$q(s),t(s)\to q(f(s)),t(f(s))$2

so naïvely testing the discretized variables answers the wrong question. The DCT framework addresses this with bridge equations that map cell probabilities or tail events back to latent covariances $q(s),t(s)\to q(f(s)),t(f(s))$3, and then infers conditional independence through the precision matrix via nodewise regression. The later DCT-GMM refinement formulates the same recovery problem as an over-identifying restrictions problem: if two discretized variables have cardinalities $q(s),t(s)\to q(f(s)),t(f(s))$4 and $q(s),t(s)\to q(f(s)),t(f(s))$5, there are $q(s),t(s)\to q(f(s)),t(f(s))$6 moment equations for a single latent covariance parameter, and one-step or two-step GMM exploits all of them rather than one binarized summary. The test then targets $q(s),t(s)\to q(f(s)),t(f(s))$7, equivalently $q(s),t(s)\to q(f(s)),t(f(s))$8, with asymptotic variance determined by the GMM Jacobian and covariance matrices (Sun et al., 2024, Sun et al., 10 Jun 2025).

A third formulation appears in functional dependence testing for discretized stochastic processes. Distance covariance is defined at the process level through the $q(s),t(s)\to q(f(s)),t(f(s))$9 norm, but in practice only grid observations are available. The weighted discretization

$\tilde H = p_t + H = 0.$0

is constructed so that Euclidean norms approximate $\tilde H = p_t + H = 0.$1 norms. Under explicit mesh conditions linking $\tilde H = p_t + H = 0.$2 to sample size and increment regularity, the discretized sample distance covariance and correlation converge to the same targets as their functional counterparts. In particular,

$\tilde H = p_t + H = 0.$3

and the discretized statistic preserves that characterization asymptotically, with bootstrap consistency for the null law of the degenerate $\tilde H = p_t + H = 0.$4-statistic (Dehling et al., 2018).

5. Bias, contradictions, and metric dependence

Several papers define discretization independence by showing where it fails. In Bayesian modeling of finite element discretization error, the central contradiction arises from the Green’s-function prior. On the fine space one obtains

$\tilde H = p_t + H = 0.$5

and the exact relation

$\tilde H = p_t + H = 0.$6

where $\tilde H = p_t + H = 0.$7 is the discretization error relative to the fine solution. Yet $\tilde H = p_t + H = 0.$8 depends on the forcing term $\tilde H = p_t + H = 0.$9, whereas the posterior covariance $q(s),t(s)$0 depends only on the stiffness matrices and the coarse/fine embedding. The paper therefore concludes that the posterior covariance “encodes the discretization error for all load cases simultaneously, yet fails to represent the discretization error for any one load case in particular.” Its proposed remedy rescales the eigenvalues of $q(s),t(s)$1 by force-dependent coefficients in the covariance eigenbasis, producing a positive semidefinite covariance whose standard deviation tracks the realized error more closely (Poot et al., 2023).

In supervised discretization for classification, the empirical study of CAIM rejects any simple claim of protocol independence. Across seven UCI datasets, sample sizes

$q(s),t(s)$2

seven classifier families, and over 117,000 models with ten replications per setup, the study finds that smaller sample size combined with CAIM produces optimistic bias, whereas discretizing inside cross-validation folds tends to create a negative bias that underestimates true performance. The paper further argues that dataset composition matters, especially the number and type of predictor attributes, with the strongest discrepancies appearing in fully numeric datasets such as Gamma, Wine Quality, and Yeast. This directly contradicts any view of discretization as a harmless preprocessing step independent of sample size or evaluation pipeline (Bennett, 2012).

Metric-sensitive formulations sharpen the same point. For discrete random variables, the independence model itself is metric-independent: it is a Segre, Veronese, or more generally Segre–Veronese variety in the probability simplex. By contrast, the Wasserstein distance to that model,

$q(s),t(s)$3

depends on the chosen metric $q(s),t(s)$4 on the state space. The induced Lipschitz polytope, the dual Wasserstein ball, the optimizer type, the piecewise algebraic formula for the distance, and even the number of nearest points can change when $q(s),t(s)$5 changes. The paper notes a special coincidence in the Hardy–Weinberg example, where the discrete and $q(s),t(s)$6 metrics give the same formula, but treats that as exceptional rather than generic. This shows that discretization independence is not automatic even when the statistical model being approximated is fixed (Çelik et al., 2020).

6. Function-space learning and signature methods

In operator learning for partial differential equations, discretization independence is elevated to a design principle. The paper distinguishes the abstract operator-learning problem from its numerical realization by discretization maps

$q(s),t(s)$7

and defines an effective discretization set

$q(s),t(s)$8

A model is “more discretization independent” if this effective set is larger. The proposed architecture is an encode-approximate-reconstruct map

$q(s),t(s)$9

with linear encoder and reconstructor implemented through learned basis functions,

$q_n,t_n$0

Because the encoder requires only quadrature of learned basis functions against the discretized input, the same numerical model can consume many meshes and fidelities; because the reconstructor returns a function rather than values on a fixed grid, the output can subsequently be sampled in many ways. Under this formulation, multifidelity learning becomes natural rather than auxiliary, and the experiments on fractional Poisson, viscous Burgers’, and Navier–Stokes equations report that multifidelity training improves accuracy and computational efficiency while further enhancing empirical discretization independence (Hauck et al., 9 Jul 2025).

A related but distinct notion appears in signature methods for time-augmented stochastic processes. Time augmentation $q_n,t_n$1 introduces the time letter $q_n,t_n$2 and thereby algebraic redundancy among truncated signature coordinates. Using the shuffle identity and the fact that

$q_n,t_n$3

the paper shows that all components of length at most $q_n,t_n$4 lie in the span of the length-$q_n,t_n$5 components. It then identifies natural subfamilies with the same span, notably words not ending in time and words not starting in time, and proves that these choices minimize computation time for backward and forward use of Chen’s relation, respectively. For stochastic processes solving SDEs with additive Brownian noise, the selected subfamilies are linearly independent as random variables. More strongly, if the path is discretized on a grid with mesh $q_n,t_n$6 and replaced by its piecewise linear interpolation, then the truncated signature converges in $q_n,t_n$7 and the same chosen family remains linearly independent for all sufficiently fine discretizations. Here discretization independence takes the form of persistence of a basis property under sampling and interpolation, not invariance of numerical values (Bourdon et al., 15 Jan 2026).

Taken together, these formulations show that discretization independence is a family resemblance rather than a single theorem. In its strongest form it is exact symmetry restoration, as in perfect path-integral composition. In discrete gravity it becomes triangulation independence and, in 4D, a diagnosis of unavoidable non-locality. In statistics it often means recovery of a latent target or asymptotic equivalence despite discretized observations. In machine learning it becomes a requirement that numerical performance not be tied to a particular mesh, sensor set, or fidelity. The literature therefore treats discretization independence not as the absence of discretization, but as disciplined control over what discretization is allowed to change.